= Positive energy theorem
{title2=$E\ge|\mathbf P|$}
Complete asymptotically flat <spin> initial data obeying the Einstein constraints and the <dominant energy condition> have future-causal ADM four-momentum, under the appropriate decay and boundary hypotheses. A spinorial proof solves the <Witten spinor equation> and integrates the <Nester two-form>. Its divergence is a sum of a nonnegative spinor-gradient term and a matter-energy contraction with a future-causal <Dirac current>. Positivity for every asymptotic constant spinor yields the full inequality, rather than merely nonnegative energy in one frame.
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